# Grothendieck's Dessins d'Enfants in a Web of Dualities. II

@article{Zhou2019GrothendiecksDD, title={Grothendieck's Dessins d'Enfants in a Web of Dualities. II}, author={Jian Zhou}, journal={arXiv: Mathematical Physics}, year={2019} }

We show that the spectral curve for Eynard-Orantin topological recursions satisfied by counting Grothendieck's dessins d'enfants are related to Narayana numbers. This suggests a connection of dessins to combinatorics of Coxeter groups, noncrossing partitions, free probability theory, and cluster algebras.

## 12 Citations

### Intersection numbers on $\overline {\mathcal M}_{g,n}$ and BKP hierarchy

- Mathematics
- 2020

In their recent inspiring paper Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar…

### BKP-Affine Coordinates and Emergent Geometry of Generalized Br\'ezin-Gross-Witten Tau-Functions

- Mathematics
- 2023

. Following Zhou’s framework, we consider the emergent geometry of the generalized Br´ezin-Gross-Witten models whose partition functions are known to be a family of tau-functions of the BKP…

### Intersection numbers on $$ {\overline{M}}_{g,n} $$ and BKP hierarchy

- MathematicsJournal of High Energy Physics
- 2021

Abstract
In their recent inspiring paper, Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide…

### On affine coordinates of the tau-function for open intersection numbers

- MathematicsNuclear Physics B
- 2021

### Orbifold Euler Characteristics of $\overline{\mathcal M}_{g,n}$

- Mathematics
- 2018

We solve the problem of the computation of the orbifold Euler characteristics of $\overline{\mathcal M}_{g,n}$ using the formalism of abstract quantum field theory we develop in an earlier work.

### Hodge–GUE Correspondence and the Discrete KdV Equation

- MathematicsCommunications in Mathematical Physics
- 2020

We prove the conjectural relationship recently proposed in Dubrovin and Yang (Commun Number Theory Phys 11:311–336, 2017) between certain special cubic Hodge integrals of the Gopakumar–Mariño–Vafa…

### Kac-Schwarz Operators of Type $B$, Quantum Spectral Curves, and Spin Hurwitz Numbers

- Mathematics
- 2022

. Given a tau-function τ ( t ) of the BKP hierarchy satisfying τ (0) = 1, we discuss how to recover its BKP-aﬃne coordinates on the isotropic Sato Grassmannian from BKP-wave function. Using this…

### BKP hierarchy, affine coordinates, and a formula for connected bosonic n-point functions

- MathematicsLetters in Mathematical Physics
- 2022

We derive a formula for the connected n-point functions of a tau-function of the BKP hierarchy in terms of its affine coordinates. This is a BKP-analogue of a formula for KP tau-functions proved by…

### Flat structure and potential vector fields related with algebraic solutions to Painlevé VI equation

- Mathematics
- 2018

A potential vector field is a solution of an extended WDVV equation which is a generalization of a WDVV equation. It is expected that potential vector fields corresponding to algebraic solutions of…

### Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals

- MathematicsAnnales Henri Poincaré
- 2020

We consider the Laguerre partition function and derive explicit generating functions for connected correlators with arbitrary integer powers of traces in terms of products of Hahn polynomials. It was…

## References

SHOWING 1-10 OF 124 REFERENCES

### Topological Recursions of Eynard–Orantin Type for Intersection Numbers on Moduli Spaces of Curves

- Mathematics
- 2013

We prove that the Virasoro constraints satisfied by the higher Weil–Petersson volumes of moduli spaces of curves are equivalent to Eynard–Orantin topological recursions for some spectral curve. This…

### The Equivariant Gromov-Witten theory of P**1

- Mathematics
- 2002

We express all equivariant Gromov-Witten invariants of the projective line as matrix elements of explicit operators acting in the Fock space. As a consequence, we prove the equivariant theory is…

### The Toda conjecture

- Mathematics
- 2001

We study the Toda conjecture of Eguchi and Yang for the Gromov-Witten invariants of CP^1,using the bihamiltonian method of the formal calculus of variations. We also study its relationship to the…

### The Grothendieck theory of dessins d'enfants: Dessins d'enfants on the Riemann sphere

- Mathematics
- 1994

In part I of this article we define the Grothendieck dessins and recall the description of the Grothendieck correspondence between dessins and Belyi pairs (X,β) where X is a compact connected Riemann…

### Geometric balance of cuspidal points realizing dessins d'enfants on the Riemann sphere

- Mathematics
- 2001

Abstract. In this paper we give necessary and sufficient conditions for a finite set of points A in
$\C\cup\{\infty\}$ to be the set of cuspidal points for a Belyi function of genus zero. The most…

### Higher genera Catalan numbers and Hirota equations for extended nonlinear Schrödinger hierarchy

- MathematicsLetters in Mathematical Physics
- 2021

We consider the Dubrovin–Frobenius manifold of rank 2 whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the…

### Gromov-Witten theory, Hurwitz theory, and completed cycles

- Mathematics
- 2002

We establish an explicit equivalence between the stationary sector of the Gromov-Witten theory of a target curve X and the enumeration of Hurwitz coverings of X in the basis of completed cycles. The…

### Enumeration of Grothendieck's dessins and KP hierarchy

- Mathematics
- 2013

Branched covers of the complex projective line ramified over $0,1$ and $\infty$ (Grothendieck's {\em dessins d'enfant}) of fixed genus and degree are effectively enumerated. More precisely, branched…

### A mathematical theory of the topological vertex

- Mathematics
- 2004

We have developed a mathematical theory of the topological vertex--a theory that was original proposed by M. Aganagic, A. Klemm, M. Marino, and C. Vafa in hep-th/0305132 on effectively computing…

### Classical Hurwitz numbers and related combinatorics

- Mathematics
- 2016

In 1891 Hurwitz [30] studied the number Hg,d of genus g ≥ 0 and degree d ≥ 1 coverings of the Riemann sphere with 2g + 2d− 2 fixed branch points and in particular found a closed formula for Hg,d for…